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dct() - Signal Processing

```y = dct(x) returns
the unitary discrete cosine transform of x,y(k)=w(k)∑n=1Nx(n)cos(π2N(2n−1)(k−1)), k=1,2,…,N,wherew(k)={1N,k=1,2N,2≤k≤N,N is the length of x,
and x and y are the same size.
If x is a matrix, dct transforms
its columns. The series is indexed from n = 1 and k = 1
instead of the usual n = 0
and k = 0 because MATLAB® vectors
run from 1 to N instead of from 0 to N – 1.y = dct(x,n) pads
or truncates x to length n before
transforming.The DCT is closely related to the discrete Fourier transform.
You can often reconstruct a sequence very accurately from only a few
DCT coefficients, a useful property for applications requiring data
reduction.```

Syntax

`y = dct(x)y = dct(x,n)`

Example

```Energy Stored in DCT CoefficientsOpen This Example
Find how many DCT coefficients represent 99% of the energy in a sequence.
x = (1:100) + 50*cos((1:100)*2*pi/40);
X = dct(x);
[XX,ind] = sort(abs(X),'descend');
i = 1;
while norm(X(ind(1:i)))/norm(X)<0.99
i = i + 1;
end
Needed = i;
Reconstruct the signal and compare to the original.X(ind(Needed+1:end)) = 0;
xx = idct(X);

plot([x;xx]')
legend('Original',['Reconstructed, N = ' int2str(Needed)], ...
'Location','SouthEast')

Related ExamplesDCT for Speech Signal Compression```